The efficiency and the demagnetization field of a general Halbach cylinder

The maximum magnetic efficiency of a general multipole Halbach cylinder of order $p$ is found as function of $p$. The efficiency is shown to decrease for increasing absolute value of $p$. The optimal ratio between the inner and outer radius, i.e. the ratio resulting in the most efficient design, is also found as function of $p$ and is shown to tend towards smaller and smaller magnet sizes. Finally, the demagnetizing field in a general $p$-Halbach cylinder is calculated, and it is shown that demagnetization is largest either at $\cos 2p\phi=1$ or $\cos 2p\phi=-1$. For the common case of a $p=1$ Halbach cylinder the maximum values of the demagnetizing field is either at $\phi = 0,\pi$ at the outer radius, where the field is always equal to the remanence, or at $\phi = \pm \pi/2$ at the inner radius, where it is the magnitude of the field in the bore. Thus to avoid demagnetization the coercivity of the magnets must be larger than these values.